Opponent begins a showdown at a non-controlled battlefield, I send in my Rengar Trophy Hunter and win combat. Do I gain a point? Do I conquer even though I was not the active player?

Yes, you will gain a conquest point for winning that showdown, and you do successfully "Conquer" the battlefield, even though you were not the active player who started the showdown.

According to RiftJudge FAQ #3657 and FAQ #10376:

• Scoring on your opponent's turn: You are eligible to score 1 point for conquering a battlefield on your opponent's turn if you win the showdown there, provided you have not already scored that specific battlefield earlier in the same turn.
• How it works: When your opponent moves a unit to an unoccupied battlefield, they initiate a non-combat showdown and are designated the Attacker. By playing Rengar, Trophy Hunter in response, you enter that battlefield and become the Defender. Once the showdown resolves and you are the only player with units remaining, you establish control of the battlefield.
• Conquering: Establishing control of a battlefield you did not yet score this turn constitutes a Conquer action (Rule 464.1). Because you established control and the battlefield was not already scored this turn, you gain 1 point.

Summary of the Process:

1. Opponent's Move: Your opponent moves to an uncontrolled battlefield. The battlefield becomes Contested (Rule 445).
2. Surprise Defense: You play Rengar, Trophy Hunter as a [Reaction] to the contested battlefield. You are now the Defender (Rule 442.1.a.2).
3. Combat: Because units from both players are present, a combat showdown begins once the chain empties and both players pass focus.
4. Victory: If you win the combat (you are the only player with units remaining at the battlefield during the Combat Cleanup step), you establish control.
5. Score: Since you established control and haven't scored that battlefield yet this turn, you successfully Conquer it and gain 1 point.

Note: This interpretation is based on the rules for establishing control and combat resolution (Rule 461.5) and the specific FAQs regarding conquering on an opponent's turn (FAQ #3657, #10376).